\section{Empirical design}
\label{sec:design}

\subsection{The estimand}

For an outcome $Y$ from Section~\ref{sec:outcomes}, define the within-event
change $\Delta Y_{i,t}(h) = Y_{i,t}(t{+}h) - Y_{i,t}(t{-}5)$. The estimand is
the difference-in-differences
\begin{equation}
\Delta\Delta Y(h) \;=\;
\mathbb{E}\big[\Delta Y(h)\,\big|\,\text{real event}\big]
\;-\;
\mathbb{E}\big[\Delta Y(h)\,\big|\,\text{matched placebo}\big].
\label{eq:dd}
\end{equation}
In potential-outcomes language, the placebo arm estimates the counterfactual
path of cross-sectional activity that the event window would have followed had
the source stock not jumped, holding fixed the stock, the minute of day, the
month, and the calm-market state. Three biases that a naive event study would
absorb are differenced out explicitly. \emph{Volatility clustering}: shocks
arrive in already-active minutes, so post-event activity is elevated even
absent transmission; the inner difference removes the level.
\emph{Deterministic intraday drift}: breadth trends over the trading day; the
placebo, matched on minute of day, follows the same drift.
\emph{Filter-induced selection}: conditioning on a calm market at $t$ selects
minutes followed by reversion toward normal activity; the placebo is drawn
under the identical filters and inherits the identical reversion.

\subsection{Placebo construction}

For each real event $(i,t)$ the matched placebo is drawn by the following
algorithm, with a fixed seed recorded in the replication package:
\begin{enumerate}
\item collect all minutes of the same month with the same minute of day
$\tau(t)$, on a different trading day, at which (a) security $i$ has an
observable return with $|z_{i,\cdot}|<2$, (b) the market-calm filter holds,
and (c) the simultaneity filter holds;
\item draw one candidate uniformly at random; compute every outcome of
Section~\ref{sec:outcomes} at the placebo minute by the identical code path;
\item label the placebo with the real event's magnitude and sign so that all
conditional cuts (by $|z|$ bin, by sign, by state) compare real and placebo
events within the same cell.
\end{enumerate}
Events with an empty candidate set (rare) contribute to the real arm only and
are dropped from difference-in-differences estimation by the pairing on
dates within cells. The match rate exceeds 95\% throughout.

\subsection{Inference}

Outcomes are correlated across events within a trading day---mechanically so,
because events of the same minute share receivers. All inference therefore
resamples \emph{days}: for $b=1,\dots,500$, draw event dates with replacement,
recompute date-level sums and counts for both arms, and form the bootstrap
distribution of the full statistic~\eqref{eq:dd}. Standard errors are
bootstrap standard deviations; the month-cluster variant (resampling
year-months) appears in the robustness table and roughly doubles the standard
error of the network differential without changing any conclusion. With
roughly 8,500 event days, the effective sample is large, and the design's
minimum detectable effect for market-wide breadth at $h=5$ is approximately
$2\times10^{-4}$ at the 5\% level with 80\% power---one additional responding
security per twenty-five events. Null results below are powered nulls.

\subsection{Identification: what is and is not claimed}

The design identifies the \emph{association} between an isolated idiosyncratic
jump and subsequent cross-sectional tail activity, purged of ambient levels,
intraday seasonality, calm-market selection, and the stock's own base rates.
It does not identify the structural channel behind the network differential:
lagged-correlation neighbors are typically industry peers, ETF co-members, and
factor-exposure twins, and the differential aggregates fundamental
information transmission (investors rationally repricing linked firms) with
trading-channel transmission (basket flows, hedging, liquidity withdrawal).
Separating those requires timestamped news and order-flow data and is beyond
OHLCV bars; the paper's claims are correspondingly phrased as predictive
propagation, not causal transmission. Likewise, the systemic co-jump series
conflates common news arrivals with endogenous synchronization; we document
its evolution and severity without attributing minutes to either source.

\subsection{Falsification battery}

Five falsifiable predictions are tested and reported. (1) \emph{No
pre-event response}: real-event outcomes at $t-5$ equal placebo levels.
(2) \emph{No placebo network structure}: the neighbor--non-neighbor
differential computed on placebo events alone is zero
($-2.7\pm2.8\times10^{-4}$). (3) \emph{No announcement artifact}: excluding
the 10:00 and 14:00--14:30 ET windows that host most scheduled releases
leaves all estimates unchanged. (4) \emph{No coverage artifact}: fixed-rule
lagged universes make the 2020 source-coverage expansion irrelevant to any
comparison. (5) \emph{No tick-regime artifact}: the post-decimalization
subsample reproduces the market-wide null and the network differential. All
five pass; Table~\ref{tab:robustness} quantifies each.